**HOW TO SUBTRACT
A DROMEDARY CAMEL FROM A BACTRIAN CAMEL
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*Let two distinguished physics professors show you!*

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Copyright 2005 by Bibhas R. De

**WELCOME!
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The Dromedary camel is one-humped, and the Bactrian camel is two-humped. Now consider this problem:
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[Source: http://www.probertencyclopaedia.com/j/Camel.jpg] [Source:http://www.peoriahs.org/2004Images/camel.jpg]

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You know the answer?! Don’t say it! Let two distinguished physics professors explain how to do the subtraction.
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Here is the problem in mathematical terms: You are given two different mathematical expressions for “the profile of a camel”. You are asked to determine if the two expressions are identical (they don’t look the same). How would you test?
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*A work of Victor Vasarely*

How patterns may emerge...

If you spot-test in the outer regions, there is no pattern

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OF CAMEL HUMPS AND MAGNETIC BUMPS
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The problem of the source-free magnetic structures I derived can be likened to the “camel problem”. I derived two “profiles” of conventional sourced magnetic structures from two different angles of physics. I then subtract one from the other, causing the source to vanish. What remains of the profiles? That is the question.

The dispute can be described with extreme mathematical precision. It concerns two cylindrically symmetric space distributions of vector quantities **A1**(r, z) [The structure ∑ _1 in the above figure] and **A2**(r, z) [The structure ∑ _2 in the above figure]. My point is that they agree in some portions of the space, but disagree in others. I present mathematical arguments for this disagreement. The difference between the two, **A **= **A1** – **A2**, is the source-free structure I derive.

So that is the problem I pitched at the Physics Establishment.

The Establishment’s point, without refuting my proof, is that **A1** and **A2** are identical at every point in space, so that there is no source-free structure (**A **= **A1** – **A2** = **0** everywhere).

When one gets past their pontifical refrain that my ideas violate some theorem or other, one gets to the crux of the true dispute. But no one was willing to go there!

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TWO WHO CAME TO BAT - RELUCTANTLY
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However, I worked the Establishment’s own review/appeals system to its fullest to get one professor to address this issue: Raymond S. Willey of the University of Pittsburg.

After the publication of the paper, Russell M. Kulsrud of Princeton University became irate that a paper he had rejected earlier was published by another journal – and revealed his identity and took up the challenge.

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PROFESSOR WILLEY AT THE BAT
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*Professor Raymond S. Willey*

Willey also presents the standard argument, but much to his credit, he recognizes that this may not be enough. He seems to see that an idea constructed from first principles cannot be dismissed from within the framework of a derived theorem - no matter how well established. Therefore he takes the time to write a computer program to compare the two distributions. He then calculates the two functions at a couple of points, and becomes convinced that the two distributions are identical. He confines his comparison to that region of space where the computations are easy (quick) to perform. The potential fallacy of this argument is obvious.

See a full description of Willey's study in his own words here.

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PROFESSOR KULSRUD AT THE BAT
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*Professor Russell Kulsrud*

Kulsrud developed a good grasp of the dispute early on and identified three facets of the dispute:

(1). Are the two distributions exactly identical?

(2). Does **A** go to zero at infinite distances?

(3). Does **A** have any singularites or discontinuities on the z = 0 plane?

Kulsrud then shows from his calculations that the answer to (2) is Yes and the answer to (3) is No. These calculations are consistent with a source-free structure. Therefore, only (1) remains to be addressed.

Kulsrud then demonstrates that for r going to zero, the two distributions approach equality. But this is exactly what I also say in my paper – there is no disagreement here. Kulsrud beautifies his point by demonstrating that, expressed as two power series expansions, the two potentials agree out to the second terms. This is nice calculation, but it does not add anything. If two functions approach each other asymptotically as r goes to zero, then if you go closer and closer to r = 0, you expect more and more terms to agree. At r = 0 all terms have to agree trivially!

See a full description of Kulsrud's work in his own words here.

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THE CAMEL ANALOGY
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Now let us work our camel analogy. The citing of the theorems – without actually addressing the problem posed - really boils down to making a blanket statement:

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We hold this truth to be self-evident that all camels are created equal…*

Willey’s batting style boils down to roughly this: Let us choose a nice, smooth region (like the camel’s rump area) and make our comparison. If the two expressions are the same here, then they are the same everywhere (if the butts are the same, the camels are the same).

Kulsrud’s batting style boils down to this: Let us choose an extremity (read tail). If the two extremities are asymptotically the same, then the two expressions are identical (If the tails are the same, the camels are the same).

This is why I say that the issue remains unresolved in spite of the best efforts of two people recognized by the Establishment as their leading experts. On the other hand, from certain things Kulsrud says, it seems to me that the inequality is emerging from his own calculations. As this began to be seen, Kulsrud terminated or suspended his study. Not having seen his calculations, I cannot comment further.

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AND NOW, THE SOURCE-FREE MAGNETIC STRUCTURE!
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[Source: http://www.probertencyclopaedia.com/j/Camel.jpg] [Source:http://www.peoriahs.org/2004Images/camel.jpg]